The scale of infinities

Property — Hierarchy of infinities

As $x\to+\infty$ the elementary functions grow at different speeds. The order is $\log x \ll x^\alpha \ll a^x \ll x! \ll x^x,\qquad (\alpha>0,\ a>1).$ The symbol $f\ll g$ means $\displaystyle\lim_{x\to+\infty} \frac{f(x)}{g(x)} = 0$: the “slower” function is negligible compared with the “faster” one.

In practice: the logarithm grows more slowly than every power; every power grows more slowly than every exponential (with base $>1$); the exponential is beaten by the factorial, and this by $x^x$. In an $\infty/\infty$ form the term highest in the scale always “wins”.

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Topics: Limits
Concepts: Hierarchy of infinities · Infinity
Methods: Limits by comparison
Skills: Computing limits